11 - Duval

Summary

Clever triviality is the essence of geniality.

E.B. Gebstadter, Copper, Silver, Gold: an Indestructible Metallic Alloy

Kronecker's Model gives a powerful tool for computing, at least within the field of the algebraic complex numbers, and for solving polynomial equations there, provided we have an algorithm for factorizing polynomials over a given algebraic extension of the rationals. Such an algorithm exists, but its practical complexity is so unsatisfactory, that the solution of polynomial equations provided by Kronecker's ideas has no practical impact and the state of the art on Solving Polynomial Equation Systems was again in an impasse: as Macaulay put it. ‘the solution is only a theoretical one’…

… until in 1987, more than one hundred years after Kronecker's Grundzüge, Duval added an unexpected twist to Kronecker's proposal, showing how factorization can be easily avoided. Her proposal threw light on Kronecker's ideas, clarifying the philosophy behind them.

I will introduce Duval's idea by discussing how to represent rings explicitly.

Explicit Representation of Rings

In all the cases we have seen up to now, a ring A is effectively given by taking a set R, whose elements are in biunivocal correspondence with the elements of A, and defining in R those operations which turn R into a ring isomorphic to A.